Problem: Rosie went on a hiking trip. The first day she walked $18$ kilometers. Each day since, she walked $90\%$ of what she walked the day before. What is the total distance Rosie has traveled by the end of the $10^\text{th}$ day? Round your final answer to the nearest kilometer.
Explanation: Notice that Rosie's daily distances form a geometric sequence. The total distance Rosie hikes after $ n$ days is the ${\text{sum}}$ of the first $n$ terms in the sequence. This is called a geometric series. This is the formula for that sum: $ S={a}\left(\dfrac{1-{r}^{ n}}{1-{r}}\right)$ where ${a}$ is the first term and ${r}$ is the common ratio. We can use this formula, along with the given information, to find the value of the sum, $ S$. Using the given information We are given that Rosie walked ${18\,\text{km}}$ on the first day. This is the first term $ a$. We are given that each daily distance is ${90\%}$ of the previous day's distance. This is the common ratio $ r$. There are ${10}$ days in the series. This is the number of terms $ n$. We want to find the total distance. This is the sum $ S$. Finding the sum $\begin{aligned} S&={18} \cdot \dfrac{1-\left({0.90}\right)^{{10}}}{1-\left({0.90}\right)} \\\\ \phantom{S}&\approx{18} \cdot \dfrac{0.6513 }{0.10} \\\\ \phantom{S}& =117.24\,\text{km} \end{aligned}$ Answer To the nearest kilometer, the total distance Rosie traveled by the end of the $10^\text{th}$ day is $117\,\text{km}$.